Rainbow triangles in three-colored graphs

Abstract

Erdős and Sós proposed a problem of determining the maximum number F (n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4 k for all k ≥ 0. These results imply that lim F (n) ( n 3 ) = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments.

Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

“…Flag algebras can be used as a general tool to attack problems from extremal combinatorics. Flag algebras were used for a wide range of problems, for example the Caccetta-Häggkvist conjecture [15,21], Turán-type problems in graphs [7,11,13,19,23,26,27], 3-graphs [9,10] and hypercubes [1,3], extremal problems in a colored environment [2,4,6], and also to problems in geometry [17] or extremal theory of permutations [5]. For more details on these applications, see a recent survey of Razborov [24].…”

“…In this paper, we use a method that we originally developed for maximizing the number of rainbow triangles in 3-edge-colored complete graphs [4]. However, the application of the method to the C 5 problem is less technical, and therefore this paper is a more accessible exposition of this new method.…”

Closing in on Hill's Conjecture

“…Our proof makes essential use of flag algebras. This powerful tool, introduced by Razborov [38], has been the basis of several recent groundbreaking results in a variety of combinatorial and geometric problems, such as [10,12,13,19,25,27,37,39], to name just a few.…”

Frustrated triangles

“…More recently, Rozborov [14] determined completely the minimum number of triangles in a graph with a given number of edges using flag algebras. Some bounds for other combinations of t 0 , t 1 , t 2 , t 3 were also given in [6,12], while similar results for three-colored graphs have recently been proved in [1,2].…”